Question

linear equations and inequalities
heather writes this system of equations to represent two lines drawn on
the coordinate plane.
\\(-6x + 18y = 6\\)
\\(4x - 12y = 20\\)
after applying the linear combination method, heather arrived at the equation
\\(0 = 60\\). what conclusion can be drawn about the system of equations?

• the equation has a solution at \\((0, 60)\\); therefore,

the system of equations has a solution at \\((0, 60)\\).

• the equation has a solution at \\((0, 0)\\); therefore,

the system of equations has a solution at \\((0, 0)\\).

• the equation has no solution; therefore, the

system of equations has no solution.

• the equation has infinite solutions; therefore, the

system of equation has infinite solutions

Explanation:

Step1: Recall system solution rules

A system of linear equations is solved by finding \(x,y\) that satisfy both. When using linear combination (elimination), if we get a contradiction like \(0 = 60\) (which is false), the system has no solution. If we get \(0 = 0\) (true), there are infinite solutions. If we get a value for \(x\) or \(y\), there's one solution.

Step2: Analyze \(0 = 60\)

The equation \(0 = 60\) is never true. This means there are no values of \(x\) and \(y\) that satisfy both original equations simultaneously. So the system has no solution.

Answer:

The equation has no solution; therefore, the system of equations has no solution.